I know of no evidence that skinny tires are more prone to flats than wider ones. However, I can understand how people might come to this conclusion:
1) Most skinny tires are intended for use by sport and recreational cyclists, who care more for light weight and low rolling resistance than reliability.
2) Skinny tires have to be pumped up more often due to their higher surface area to volume ratio. If not pumped up often enough, there may be pinch flats.
Skinny tires need not be fragile, lightweight things. For example, the Schwalbe Marathon Plus is available in a 25-622 size and, like all of that model, is about as flat resistant as a car tire.
Paul